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In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial.According to the theorem, the power (+) expands into a polynomial with terms of the form , where the exponents and are nonnegative integers satisfying + = and the coefficient of each term is a specific positive integer ...
Binomial expansion. Add languages. Add links. Article; ... Download as PDF; Printable version; In other projects Appearance. move to sidebar hide. From Wikipedia, the ...
Pascal's pyramid is the three-dimensional analog of the two-dimensional Pascal's triangle, which contains the binomial numbers and relates to the binomial expansion and the binomial distribution. The binomial and trinomial numbers, coefficients, expansions, and distributions are subsets of the multinomial constructs with the same names.
Download as PDF; Printable version; In other projects Wikidata item; ... is a mathematical identity involving sums of binomial coefficients. It states the following:
The third power of the trinomial a + b + c is given by (+ +) = + + + + + + + + +. This can be computed by hand using the distributive property of multiplication over addition and combining like terms, but it can also be done (perhaps more easily) with the multinomial theorem.
The generalized binomial theorem gives (+) = = = + + +.A proof for this identity can be obtained by showing that it satisfies the differential equation ...
Lucas's theorem can be generalized to give an expression for the remainder when () is divided by a prime power p k.However, the formulas become more complicated. If the modulo is the square of a prime p, the following congruence relation holds for all 0 ≤ s ≤ r ≤ p − 1, a ≥ 0, and b ≥ 0.
In mathematics, Pascal's rule (or Pascal's formula) is a combinatorial identity about binomial coefficients.It states that for positive natural numbers n and k, + = (), where () is a binomial coefficient; one interpretation of the coefficient of the x k term in the expansion of (1 + x) n.